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Horseshoe Lemma


HorseshoeLemmaDiagram

Given a short exact sequence of modules

 0->A->B->C->0,
(1)

let

 ...->P_2->^(d_2)P_1->^(d_1)P_0->^(d_0)A->0
(2)
 ...->Q_2->^(f_2)Q_1->^(f_1)Q_0->^(f_0)C->0
(3)

be projective resolutions of A and C, respectively. Then there is a projective resolution of B

 ...->P_2 direct sum Q_2->^(e_2)P_1 direct sum Q_1->^(e_1)P_0 direct sum Q_0->^(e_0)B->0
(4)

such that the above diagrams are commutative. Here, i_n is the injection of the first summand, whereas p_n is the projection onto the second factor for n>=0.

HorseshoeLemmaResolutions

The name of this lemma derives from the shape of the diagram formed by the short exact sequence and the given projective resolutions.


See also

Commutative Diagram

This entry contributed by Margherita Barile

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References

Weibel, C. A. An Introduction to Homological Algebra. Cambridge, England: Cambridge University Press, pp. 37-38, 1994.

Referenced on Wolfram|Alpha

Horseshoe Lemma

Cite this as:

Barile, Margherita. "Horseshoe Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HorseshoeLemma.html

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